Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(19,26)(22,25)(27,32)(29,31);(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,26)(18,22,24,25)(20,27,28,32)(23,29,30,31);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32);(1,3)(2,6)(4,7)(5,8)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=g,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52676, #8092119, #8092476, … |
| maximal normal subgroup of | C2×#52676, #8092119, #8092476, … |
| maximal quotient of | #582328, #582513, #582545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45664);